C# C语言中的数学优化#
我整天都在分析一个应用程序,在优化了几段代码之后,我的待办事项列表上就剩下这个了。这是一个神经网络的激活函数,它被调用超过1亿次。根据dotTrace,它大约占整个功能时间的60% 您将如何对此进行优化C# C语言中的数学优化#,c#,optimization,neural-network,performance,C#,Optimization,Neural Network,Performance,我整天都在分析一个应用程序,在优化了几段代码之后,我的待办事项列表上就剩下这个了。这是一个神经网络的激活函数,它被调用超过1亿次。根据dotTrace,它大约占整个功能时间的60% 您将如何对此进行优化 public static float Sigmoid(double value) { return (float) (1.0 / (1.0 + Math.Pow(Math.E, -value))); } 想法:也许您可以使用预先计算的值制作一个(大)查找表?试试: public st
public static float Sigmoid(double value) {
return (float) (1.0 / (1.0 + Math.Pow(Math.E, -value)));
}
想法:也许您可以使用预先计算的值制作一个(大)查找表?试试:
public static float Sigmoid(double value) {
return 1.0f / (1.0f + (float) Math.Exp(-value));
}
编辑:我做了一个快速基准测试。在我的机器上,上面的代码比你的方法快43%,这个数学上等价的代码是最快的(比原始代码快46%):
编辑2:我不确定C#函数的开销有多大,但如果在源代码中包含,应该可以使用它,它使用float exp函数。可能会快一点
public static float Sigmoid(double value) {
float k = expf((float) value);
return k / (1.0f + k);
}
此外,如果您正在进行数百万次调用,那么函数调用开销可能是一个问题。尝试创建一个内联函数,看看是否有帮助。在1亿次调用中,我开始怀疑分析器开销是否会影响结果。将计算替换为无操作,然后查看是否仍报告它消耗了60%的执行时间
或者更好的方法是,创建一些测试数据,并使用秒表计时器分析大约一百万个调用 在谷歌搜索中,我找到了Sigmoid函数的替代实现
public double Sigmoid(double x)
{
return 2 / (1 + Math.Exp(-2 * x)) - 1;
}
这符合你的需要吗?速度快吗
如果是针对激活函数,那么如果e^x的计算完全准确,这是否非常重要 例如,如果使用近似值(1+x/256)^256,在我的奔腾Java测试中(我假设C基本上编译为相同的处理器指令),这比e^x(Math.exp())快7-8倍,精确到小数点后2位,最多为+/-1.5的x,在你所说的范围内的正确数量级内。(显然,要提高到256,实际上需要将数字平方8次——不要使用Math.Pow来实现这一点!)在Java中: 根据您希望的近似精度,将256倍或减半(以及添加/删除乘法)。即使n=4,对于-0.5和0.5之间的x值,它仍然给出了大约1.5个小数位的精度(看起来比Math.exp()快15倍)
顺便说一下,我忘了提到-,你显然不应该真的除以256:乘以常数1/256。Java的JIT编译器会自动进行这种优化(至少Hotspot是这样),我假设C也必须这样做。首先想到的是:关于values变量的一些统计数据如何
- “value”的值是否通常很小-10(使用性能测量更新)(使用实际结果再次更新:)
我认为,在性能方面,查找表解决方案可以让您走得更远,而内存和精度方面的成本可以忽略不计
下面的代码片段是C语言的一个示例实现(我说C语言不够流利,无法对其进行编译)。它运行和性能都很好,但我确信其中有一个bug:)
待办事项: 有需要改进的地方和消除弱点的方法;如何做是留给读者作为练习:)- 调整函数的范围,以避免表开始和结束的跳转
- 添加一个轻微的噪波函数以隐藏锯齿瑕疵
- 正如Rex所说,插值可以使您获得更高的精度,同时具有更便宜的性能
- Soprano对您的呼叫进行了一些很好的优化:
如果您尝试查找表,发现它使用了太多内存,您可以始终查看每个连续调用的参数值,并使用一些缓存技术 例如,尝试缓存最后一个值和结果。如果下一个调用的值与上一个调用的值相同,则不需要计算它,因为您已经缓存了最后一个结果。如果当前调用与上一次调用相同,即使是100次调用中的1次,您可能会节省100万次计算 或者,您可能会发现,在连续10次调用中,value参数平均相同2次,因此您可以尝试缓存最后10个值/答案。1)是否仅从一个位置调用此参数?如果是这样,您可以通过将代码移出该函数并将其放在通常调用Sigmoid函数的位置来获得少量性能。在代码可读性和组织方面,我不喜欢这种想法,但当您需要获得最后的性能提升时,这可能会有所帮助,因为我认为函数调用需要在堆栈上推/弹出寄存器,如果您的代码都是内联的,这是可以避免的public static float Sigmoid(double value) { float k = Math.Exp(value); return k / (1.0f + k); }
2) 我不知道这是否有帮助,但请尝试将函数参数设置为ref参数。看看是不是快一点。我本来建议让它成为const(如果C++是这样的话,这将是一个优化),但是C ^不支持const参数。 你也可以考虑用一些替代的激活函数来进行实验,这些函数比较便宜。例如:
(这可以作为f(x) = (3x - x**3)/2
少乘一次)。该函数具有奇对称性,其导数是平凡的。将其用于神经网络需要通过除以输入总数(将域限制为[-1..1],这也是范围)来规范化输入的总和f(x) = x*(3 - x*x)/2
- 请记住,此激活函数中的任何更改都是以不同行为为代价的。这甚至包括切换到浮动(从而降低精度)或使用激活替代物。只有对您的用例进行实验,才能找到正确的方法 <> LI>除了简单的代码优化之外,我还建议考虑< <强> >计算的强>并行(即:利用多个核O)
if(sigmoidCache.containsKey(value)) return sigmoidCache.get(value);
#include <math.h>
#include <stdio.h>
#include <time.h>
#define SCALE 320.0f
#define RESOLUTION 2047
#define MIN -RESOLUTION / SCALE
#define MAX RESOLUTION / SCALE
static float sigmoid_lut[RESOLUTION + 1];
void init_sigmoid_lut(void) {
int i;
for (i = 0; i < RESOLUTION + 1; i++) {
sigmoid_lut[i] = (1.0 / (1.0 + exp(-i / SCALE)));
}
}
static float sigmoid1(const float value) {
return (1.0f / (1.0f + expf(-value)));
}
static float sigmoid2(const float value) {
if (value <= MIN) return 0.0f;
if (value >= MAX) return 1.0f;
if (value >= 0) return sigmoid_lut[(int)(value * SCALE + 0.5f)];
return 1.0f-sigmoid_lut[(int)(-value * SCALE + 0.5f)];
}
float test_error() {
float x;
float emax = 0.0;
for (x = -10.0f; x < 10.0f; x+=0.00001f) {
float v0 = sigmoid1(x);
float v1 = sigmoid2(x);
float error = fabsf(v1 - v0);
if (error > emax) { emax = error; }
}
return emax;
}
int sigmoid1_perf() {
clock_t t0, t1;
int i;
float x, y = 0.0f;
t0 = clock();
for (i = 0; i < 10; i++) {
for (x = -5.0f; x <= 5.0f; x+=0.00001f) {
y = sigmoid1(x);
}
}
t1 = clock();
printf("", y); /* To avoid sigmoidX() calls being optimized away */
return (t1 - t0) / (CLOCKS_PER_SEC / 1000);
}
int sigmoid2_perf() {
clock_t t0, t1;
int i;
float x, y = 0.0f;
t0 = clock();
for (i = 0; i < 10; i++) {
for (x = -5.0f; x <= 5.0f; x+=0.00001f) {
y = sigmoid2(x);
}
}
t1 = clock();
printf("", y); /* To avoid sigmoidX() calls being optimized away */
return (t1 - t0) / (CLOCKS_PER_SEC / 1000);
}
int main(void) {
init_sigmoid_lut();
printf("Max deviation is %0.6f\n", test_error());
printf("10^7 iterations using sigmoid1: %d ms\n", sigmoid1_perf());
printf("10^7 iterations using sigmoid2: %d ms\n", sigmoid2_perf());
return 0;
}
$ gcc -O2 test.c -o test && ./test
Max deviation is 0.001664
10^7 iterations using sigmoid1: 571 ms
10^7 iterations using sigmoid2: 113 ms
public static float Sigmoid(double value)
{
float k = Math.Exp(value);
return k / (1.0f + k);
}
f(x) = (3x - x**3)/2
f(x) = x*(3 - x*x)/2
void sigmoid_sse(float *a_Values, float *a_Output, size_t a_Size){
__m128* l_Output = (__m128*)a_Output;
__m128* l_Start = (__m128*)a_Values;
__m128* l_End = (__m128*)(a_Values + a_Size);
const __m128 l_One = _mm_set_ps1(1.f);
const __m128 l_Half = _mm_set_ps1(1.f / 2.f);
const __m128 l_OneOver6 = _mm_set_ps1(1.f / 6.f);
const __m128 l_OneOver24 = _mm_set_ps1(1.f / 24.f);
const __m128 l_OneOver120 = _mm_set_ps1(1.f / 120.f);
const __m128 l_OneOver720 = _mm_set_ps1(1.f / 720.f);
const __m128 l_MinOne = _mm_set_ps1(-1.f);
for(__m128 *i = l_Start; i < l_End; i++){
// 1.0 / (1.0 + Math.Pow(Math.E, -value))
// 1.0 / (1.0 + Math.Exp(-value))
// value = *i so we need -value
__m128 value = _mm_mul_ps(l_MinOne, *i);
// exp expressed as inifite series 1 + x + (x ^ 2 / 2!) + (x ^ 3 / 3!) ...
__m128 x = value;
// result in l_Exp
__m128 l_Exp = l_One; // = 1
l_Exp = _mm_add_ps(l_Exp, x); // += x
x = _mm_mul_ps(x, x); // = x ^ 2
l_Exp = _mm_add_ps(l_Exp, _mm_mul_ps(l_Half, x)); // += (x ^ 2 * (1 / 2))
x = _mm_mul_ps(value, x); // = x ^ 3
l_Exp = _mm_add_ps(l_Exp, _mm_mul_ps(l_OneOver6, x)); // += (x ^ 3 * (1 / 6))
x = _mm_mul_ps(value, x); // = x ^ 4
l_Exp = _mm_add_ps(l_Exp, _mm_mul_ps(l_OneOver24, x)); // += (x ^ 4 * (1 / 24))
#ifdef MORE_ACCURATE
x = _mm_mul_ps(value, x); // = x ^ 5
l_Exp = _mm_add_ps(l_Exp, _mm_mul_ps(l_OneOver120, x)); // += (x ^ 5 * (1 / 120))
x = _mm_mul_ps(value, x); // = x ^ 6
l_Exp = _mm_add_ps(l_Exp, _mm_mul_ps(l_OneOver720, x)); // += (x ^ 6 * (1 / 720))
#endif
// we've calculated exp of -i
// now we only need to do the '1.0 / (1.0 + ...' part
*l_Output++ = _mm_rcp_ps(_mm_add_ps(l_One, l_Exp));
}
}
public static float Sigmoid(double value) {
float v = value;
float k = Math.Exp(v);
return k / (1.0f + k);
}
public static double Exp(double val) {
long tmp = (long) (1512775 * val + 1072632447);
return BitConverter.Int64BitsToDouble(tmp << 32);
}
$ gmcs -optimize test.cs && mono test.exe
Max deviation is 0.001663983
10^7 iterations using Sigmoid1() took 1646.613 ms
10^7 iterations using Sigmoid2() took 237.352 ms
using System;
using System.Diagnostics;
class LUTTest {
private const float SCALE = 320.0f;
private const int RESOLUTION = 2047;
private const float MIN = -RESOLUTION / SCALE;
private const float MAX = RESOLUTION / SCALE;
private static readonly float[] lut = InitLUT();
private static float[] InitLUT() {
var lut = new float[RESOLUTION + 1];
for (int i = 0; i < RESOLUTION + 1; i++) {
lut[i] = (float)(1.0 / (1.0 + Math.Exp(-i / SCALE)));
}
return lut;
}
public static float Sigmoid1(double value) {
return (float) (1.0 / (1.0 + Math.Exp(-value)));
}
public static float Sigmoid2(float value) {
if (value <= MIN) return 0.0f;
if (value >= MAX) return 1.0f;
if (value >= 0) return lut[(int)(value * SCALE + 0.5f)];
return 1.0f - lut[(int)(-value * SCALE + 0.5f)];
}
public static float error(float v0, float v1) {
return Math.Abs(v1 - v0);
}
public static float TestError() {
float emax = 0.0f;
for (float x = -10.0f; x < 10.0f; x+= 0.00001f) {
float v0 = Sigmoid1(x);
float v1 = Sigmoid2(x);
float e = error(v0, v1);
if (e > emax) emax = e;
}
return emax;
}
public static double TestPerformancePlain() {
Stopwatch sw = new Stopwatch();
sw.Start();
for (int i = 0; i < 10; i++) {
for (float x = -5.0f; x < 5.0f; x+= 0.00001f) {
Sigmoid1(x);
}
}
sw.Stop();
return sw.Elapsed.TotalMilliseconds;
}
public static double TestPerformanceLUT() {
Stopwatch sw = new Stopwatch();
sw.Start();
for (int i = 0; i < 10; i++) {
for (float x = -5.0f; x < 5.0f; x+= 0.00001f) {
Sigmoid2(x);
}
}
sw.Stop();
return sw.Elapsed.TotalMilliseconds;
}
static void Main() {
Console.WriteLine("Max deviation is {0}", TestError());
Console.WriteLine("10^7 iterations using Sigmoid1() took {0} ms", TestPerformancePlain());
Console.WriteLine("10^7 iterations using Sigmoid2() took {0} ms", TestPerformanceLUT());
}
}
#light
let Scale = 320.0f;
let Resolution = 2047;
let Min = -single(Resolution)/Scale;
let Max = single(Resolution)/Scale;
let range step a b =
let count = int((b-a)/step);
seq { for i in 0 .. count -> single(i)*step + a };
let lut = [|
for x in 0 .. Resolution ->
single(1.0/(1.0 + exp(-double(x)/double(Scale))))
|]
let sigmoid1 value = 1.0f/(1.0f + exp(-value));
let sigmoid2 v =
if (v <= Min) then 0.0f;
elif (v>= Max) then 1.0f;
else
let f = v * Scale;
if (v>0.0f) then lut.[int (f + 0.5f)]
else 1.0f - lut.[int(0.5f - f)];
let getError f =
let test = range 0.00001f -10.0f 10.0f;
let errors = seq {
for v in test ->
abs(sigmoid1(single(v)) - f(single(v)))
}
Seq.max errors;
open System.Diagnostics;
let test f =
let sw = Stopwatch.StartNew();
let mutable m = 0.0f;
let result =
for t in 1 .. 10 do
for x in 1 .. 1000000 do
m <- f(single(x)/100000.0f-5.0f);
sw.Elapsed.TotalMilliseconds;
printf "Max deviation is %f\n" (getError sigmoid2)
printf "10^7 iterations using sigmoid1: %f ms\n" (test sigmoid1)
printf "10^7 iterations using sigmoid2: %f ms\n" (test sigmoid2)
let c = System.Console.ReadKey(true);
Max deviation is 0.001664
10^7 iterations using sigmoid1: 588.843700 ms
10^7 iterations using sigmoid2: 156.626700 ms
Max deviation is 0.001664
10^7 iterations using sigmoid1: 628 ms
10^7 iterations using sigmoid2: 157 ms
$ javac LUTTest.java && java LUTTest
Max deviation is 0.001664
10^7 iterations using sigmoid1() took 1398 ms
10^7 iterations using sigmoid2() took 177 ms
public class LUTTest {
private static final float SCALE = 320.0f;
private static final int RESOLUTION = 2047;
private static final float MIN = -RESOLUTION / SCALE;
private static final float MAX = RESOLUTION / SCALE;
private static final float[] lut = initLUT();
private static float[] initLUT() {
float[] lut = new float[RESOLUTION + 1];
for (int i = 0; i < RESOLUTION + 1; i++) {
lut[i] = (float)(1.0 / (1.0 + Math.exp(-i / SCALE)));
}
return lut;
}
public static float sigmoid1(double value) {
return (float) (1.0 / (1.0 + Math.exp(-value)));
}
public static float sigmoid2(float value) {
if (value <= MIN) return 0.0f;
if (value >= MAX) return 1.0f;
if (value >= 0) return lut[(int)(value * SCALE + 0.5f)];
return 1.0f - lut[(int)(-value * SCALE + 0.5f)];
}
public static float error(float v0, float v1) {
return Math.abs(v1 - v0);
}
public static float testError() {
float emax = 0.0f;
for (float x = -10.0f; x < 10.0f; x+= 0.00001f) {
float v0 = sigmoid1(x);
float v1 = sigmoid2(x);
float e = error(v0, v1);
if (e > emax) emax = e;
}
return emax;
}
public static long sigmoid1Perf() {
float y = 0.0f;
long t0 = System.currentTimeMillis();
for (int i = 0; i < 10; i++) {
for (float x = -5.0f; x < 5.0f; x+= 0.00001f) {
y = sigmoid1(x);
}
}
long t1 = System.currentTimeMillis();
System.out.printf("",y);
return t1 - t0;
}
public static long sigmoid2Perf() {
float y = 0.0f;
long t0 = System.currentTimeMillis();
for (int i = 0; i < 10; i++) {
for (float x = -5.0f; x < 5.0f; x+= 0.00001f) {
y = sigmoid2(x);
}
}
long t1 = System.currentTimeMillis();
System.out.printf("",y);
return t1 - t0;
}
public static void main(String[] args) {
System.out.printf("Max deviation is %f\n", testError());
System.out.printf("10^7 iterations using sigmoid1() took %d ms\n", sigmoid1Perf());
System.out.printf("10^7 iterations using sigmoid2() took %d ms\n", sigmoid2Perf());
}
}
Empty Function: 79ms 0
Original: 1576ms 0.7202294
Simplified: (soprano) 681ms 0.7202294
Approximate: (Neil) 441ms 0.7198783
Bit Manip: (martinus) 836ms 0.72318
Taylor: (Rex Logan) 261ms 0.7202305
Lookup: (Henrik) 182ms 0.7204863
public static object[] Time(Func<double, float> f) {
var testvalue = 0.9456;
var sw = new Stopwatch();
sw.Start();
for (int i = 0; i < 1e7; i++)
f(testvalue);
return new object[] { sw.ElapsedMilliseconds, f(testvalue) };
}
public static void Main(string[] args) {
Console.WriteLine("Empty: {0,10}ms {1}", Time(Empty));
Console.WriteLine("Original: {0,10}ms {1}", Time(Original));
Console.WriteLine("Simplified: {0,10}ms {1}", Time(Simplified));
Console.WriteLine("Approximate: {0,10}ms {1}", Time(ExpApproximation));
Console.WriteLine("Bit Manip: {0,10}ms {1}", Time(BitBashing));
Console.WriteLine("Taylor: {0,10}ms {1}", Time(TaylorExpansion));
Console.WriteLine("Lookup: {0,10}ms {1}", Time(LUT));
}
public static double Sigmoid(double value)
{
return 0.5d + 0.5d * Math.Tanh(value/2);
}
public double Sigmoid(double value)
{
if (value < -45.0) return 0.0;
if (value > 45.0) return 1.0;
return 1.0 / (1.0 + Math.Exp(-value));
}